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Lac promoter synthesis rate
van Hoek F, Hogeweg P (2007) The effect of stochasticity on the Lac Operon: An evolutionary perspective. PLoS Comput Biol 3 (6): E111
The article's objective, as you can infer from the title, is to evaluate the effect of stochasticity in the evolution of a Promoter. In order to do so they built a comprehensive model including every parameter involved in Transcription and translation. They measure some parameters but they depend mostly on literature to define them.
They both do a deterministic and a Stochastic analysis. To generate a Stochastic model they added one parameter, the average burst size of protein translation (protein translation occurs in bursts, after a several mRNA is synthesized proteins can be translated from the same mRNA). This was possible because when an mRNA molecule is translated it can not be degraded. Therefore after each translation it can either be translated again (p) or be degraded (1-p). This suggests that protein production occurs in bursts with a burst size geometrically distributed. After they compared their noise levels in the model with the experimental noise measurements they found good agreement.
Transcription they used to model a two-dimensional Hill-function dependent on the cAMP concentration and alloctase. (repressa the glucose and lactose Operon via cAMP activates the Operon via allolactose).
They use 11 biochemical parameters, including three of special importance for us:
a, when the rate Transcription RNA Polymerase is bound to the DNA, but CRP and Laci are not. evolva: initial value: 1.1 × 10-7 mM / min
b, The Transcription rate when both RNA Polymerase and CRP are bound, but Laci is not bound to the DNA. evolva: initial value: 2.2 × 10-5 mM / min
c,''leakiness,''the Transcription rate when RNA Polymerase is not bound to the DNA. evolva: initial value 5.5 × 10-10 mM / min
They modele binomially protein degradation, assuming that cell when to divide proteins are divided randomly between the cells. However in a population of non-Dividing cells this "dilution" can not be taken in account.
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Exploring Sensibility Analysis
Normalizing sensitivity!
- d x (t) / d t
- 1 / x (t) * dx (t) / dt
- k / x (t) * d x (t) / d t
* It allows you to compare dimensionless.
How do we measure?, Do they move the k value in a range?, Or is it a property of the system?
Wilkinson, 1978.
The parameters are systematically perturbed from their given values…
… change from the given value… (although it is recommended to define each system).
Ingalls & Sauro, 2002
- Before the analysis, it is recommended that you detect the 'preserved structures' (linear units, eg moieties).
REDUCING THE SYSTEM
How?
- The response coefficient defined above provides a measure of the difference between this ‘‘perturbed trajectory’’ and the ‘‘nominal’’ (unperturbed) trajectory at each time t: As time tends to infinity, each trajectory will converge to its steady state, and so the response coefficient will converge to the steady-state response of MCA.
- At steady state, these coefficients reduce to their standard MCA counterparts—flux responses.
NOTE: The sensitivity analysis is sensitive to the initial concentrations of metabolites.
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Advances
EXPERIMENTAL
* Electrodes and measurement method.
There are, broadly speaking, four options to choose from:
1) The faculty of physiology at UNAM has a sensor for variations of voltages of orders that could be useful (Question: If it is not specific for nickel ¿Is there a way to discriminate the noise?).
Among the benefits versus the other possibilities we have that sensitivity appears to be very good and we know this because similar experiments have been done previously. This in turn gives us the assurance that the sensor has already been tested in other biological systems in line with the results expected. And plus, a member of our team already knows how to use this system. On the other hand, the fact that the Insitute is part of the UNAM has the advantage of working with people from the same team, not counting the enormous advantage of being less than 9 hours away.
2) At the University of Guadalajara, there is a device that measures the medium resistivity in the orders of 10 ^ -9 Molar. We have not checked in sufficient detail the operation of this system, but at least the sensitivity offered is very promising. Furthermore, we believe that the metal used in a phase of measurement reacts specifically with nickel also producing a easily measurable and identifiable optical effect.
Among the advantages this option provides are, that it is sensitive and that the software used to process and record each measurement is very comprehensive and drops the noise reliably. The main disadvantage, is that the apparatus is lended from the laboratory of the UdG, which means that we would have to carry biological material.
3) Someone offered to buy the specific sensor for nickel and lend it to us during the measurements phase. We need to contact him and describe the project and what we need at the time of sensing. The only requirement is that he appears as a collaborator in the experimental publication resulting from this research.
4) Finally, most certainly not least, Trejo is still arming the sensor as we had planned initially. He has progressed well and in about two weeks it will be ready.
It was decided to wait for Option 4 and, as a backup, the support of Dr. Pena (1), but this does not rule out the option 3, which will be investigated for further details such as shipping time and specificity of the device.
* We need to define the requirements for the bioparts and make the oligos. The oligos are being designed and probably by Friday they will be sent.
* Design of experiments to estimate parameters (tentative date September 10-12).
They are still working on the buildings, but there will be a first meeting on Tuesday, Sept. 23, at 4:00 pm.
WIKI
* Update the notebook.
* Update the section of the model.
We will wait to define the line that will follow the model and its simulation.
* We need to solve the problem of space.
* Correct the image format.
MODEL
* Simulation.
* Pending data.
* Analysis (... stochastic processes?).
We are working on it... We've had some problems with the simulation, and we are doing sensitivity analysis and parameter's scanning.
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Converting units
1. Reaction 1.
3.723mM =? Molecules
How to convert from Molar to Mole?
M = mole / liter
-> What is the volume (in liters) of a cell?
<<The volume of a bacterium is 10^-15L>> Molar concentration. Example. Wikipedia.
3.723mM = 37.23x10-18 mol at 10-15 liters
37.23 x10-18 mol = 224.20427x105 molecules
* 1 mol = 6.02214x1023 molecules
2. Reaction 6.
The flow in 20 plasmids is 20mM / h ...
-> in 10 plasmids, 10mM / h
Flow = 10 mM / h = 10mM/3600s = 0.00278mM / s
but ... 0.00278mM = 0.0278x10-18mol in 10-15 liters
... 0.0278x10-18mol = 1.67415x105 molecules.
The flow in the cell is 1.67415x104 molecules / s with 10 copies (plasmids).
ν = k * [promoter]
1.67415x104 molecules / s = k * 10 molecules
-> k = 1.67415x103 molecules / s
3. Reaction 5.
ΔG ° =- 23.81 kcal / mol
Keq = exp (-ΔG º / RT)
-> The units do not affect using this formula!
Is it correct?
Correction of the synthesis reaction of cI
Units of the k3ON, estimated in reference 3 are molecules-1seconds-1, which means that the reaction is second order.
In that same article, they suggest that the mere presence of the dimer ensures the production of cI with k3ON rate (that is, that bonding is efficient).
Since the estimated values do not consider the intermediate step of the promoter's union and the complex, we should not consider it.
3. ...
3.1 ρcI + (AHL: LuxR): (AHL: LuxR) -> CI + ρcI + (AHL: LuxR): (AHL: LuxR)
k3ON
Unknown parameters
| cI Dimerization |
k4.1 & k-4.1 |
| Suppression by CI |
V5max or k5 |
| Nickel Extrusion |
k7 |
| RcnA Degradation |
k8 |
| Nickel Internalization |
k9
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Dimerization of cI
k4.1 & k-4.1?
2 cI <-> cI: cI
¿Quasi-balance?
The initial concentration of cI varies over time, but the proportion is preserved.
... Avoid kinetic analysis of the fast reactions, ie to take into consideration only their equilibrium constants instead of considering their rates (Kholodenko et al. 1998).
Keq = [cI: cI] / [cI] [cI] -> [cI:cI] = Keq [cI] 2
d (cI:cI 2cI) / dt = v+ - v -
d (Keq [cI]^2 + 2cI) / dt = v+ - v-
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General checkup
Simulation: The previous crisis was overcome... We are walking fine, however, there are still some parameters missing, but it has been decided that they will be estimated by adjusting the model and experimentally when we have the opportunity.
Data: There are a few missing parameters and we know that most of them are not available, so we have given up the search... What has yet to be defined in terms of data are concentrations of AiiA and LuxR, considering that they are preserved constants in the model.
We will also design experiments to obtain the missing parameters through the experimental measurements (viable).
Analysis: Stechiometric matrix: There was a meeting today in the morning with Osbaldo to review its analysis, we will share the information as soon as possible.
Sensitivity analysis: Although we don't understand yet the particular units in which SimBiology returns the results, the first graphics that show the basic parameters of the model are ready and they are the ones involved in the degradation of AHL and dimerization of it with LuxR ( the start of the cascade) and the ones regarding the entry and exit of Nickel. We must do this analysis recurrently, as we have seen that this is sensitive to the initial concentrations of metabolites, which are not yet fully defined.
States and stationary Jacobian: They were stopped briefly because we need the parameters for further analysis.
Experimental
Requirements: Urgent! We need to send the oligos required to synthesis; We are working on it, but let's consider this a priority.
Electrodes: The device is not ready.
Design of experiments: The first meeting will be today.
Funds & Jambouree: We are still waiting for some sponsors to reply.
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Correcting reaction 5
The Keq is formally dimensionless, but for formallity to the reaction, it is sometimes necessary to add units.
In the case of the reaction 5, the only information we have is the ΔGº of the reaction, so the Keq is calculated as exp (-ΔG º / RT), which results in a value without units.
As the definition of Keq is given in concentration, we interpret this value in terms of molarity. We know that once defined, they are completely interconvertibles: Molar <-> moles <-> molecules and since we are working on molecules for our model, we make the relevant adjustments.
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